Techniques are disclosed that implement algorithms for rapidly finding the leave-one-out (LOO) error for regularized least squares (RLS) problems over a large number of values of the regularization parameter .lamda.. Algorithms implementing the techniques use approximately the same time and space as training a single regularized least squares classifier/regression algorithm. The techniques include a classification/regression process suitable for moderate sized datasets, based on an eigendecomposition of the unregularized kernel matrix. This process is applied to a number of benchmark datasets, to show empirically that accurate classification/regression can be performed using a Gaussian kernel with surprisingly large values of the bandwidth parameter .sigma.. It is further demonstrated how to exploit this large .sigma. regime to obtain a linear-time algorithm, suitable for large datasets, that computes LOO values and sweeps over .lamda..